Newton's Method for Solving Nonlinear Equations
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Resource Overview
An iterative scheme constructed using Taylor expansion of f(x)=0, approximating the curve with linear segments where implementation requires calculating Jacobian matrices and solving linear systems at each iteration
Detailed Documentation
When constructing an iterative scheme using Taylor expansion of f(x)=0, we perform Taylor expansion of f(x)=0 around the initial value x0, then approximate the curve by representing the linear part of the expansion as straight lines in the interval [x, x+h]. This yields an approximation f(x)≈0 that becomes more accurate as h approaches zero. In practical implementation, this involves computing the Jacobian matrix at each iteration point and solving a linear system to update the solution approximation. This method provides a systematic approach for constructing iterative formats, enabling more efficient problem-solving through successive linear approximations that converge quadratically under proper conditions.
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